Properties

Label 21294cm
Number of curves $6$
Conductor $21294$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("21294.bu1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 21294cm

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
21294.bu5 21294cm1 [1, -1, 1, -6116, -404985] [2] 61440 \(\Gamma_0(N)\)-optimal
21294.bu4 21294cm2 [1, -1, 1, -127796, -17537529] [2, 2] 122880  
21294.bu3 21294cm3 [1, -1, 1, -158216, -8533209] [2, 2] 245760  
21294.bu1 21294cm4 [1, -1, 1, -2044256, -1124484825] [2] 245760  
21294.bu6 21294cm5 [1, -1, 1, 587074, -66367713] [2] 491520  
21294.bu2 21294cm6 [1, -1, 1, -1390226, 625212735] [2] 491520  

Rank

sage: E.rank()
 

The elliptic curves in class 21294cm have rank \(0\).

Modular form 21294.2.a.bu

sage: E.q_eigenform(10)
 
\( q + q^{2} + q^{4} - 2q^{5} + q^{7} + q^{8} - 2q^{10} - 4q^{11} + q^{14} + q^{16} - 2q^{17} + 4q^{19} + O(q^{20}) \)

Isogeny matrix

sage: E.isogeny_class().matrix()
 

The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with Cremona labels.